Geology Reference
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Table 7. Optimal distribution of VE dampers
the classical and fractional rheological models,
is presented. The resulting matrix equation of
motion is the fractional differential equation for
the models with fractional derivative or the clas-
sical differential equation when the dampers are
modelled using the classical rheological models.
The dynamics properties of structures are deter-
mined as the solution to the appropriately defined
linear or non-linear eigenvalue problems and as
the solution to the appropriately defined set of
algebraic equations.
The optimal damper distributions in build-
ings are found for various objective functions.
The weighted sum of amplitudes of the transfer
functions of interstorey drifts and the weighted
sum of amplitudes of the transfer functions of
displacements evaluated at the fundamental natu-
ral frequency of the frame with the dampers are
most frequently used as the objective function.
The optimization problem is solved using the
sequential optimization method and the particle
swarm optimization method. Several numerical
solutions to the considered optimization problem
are presented and discussed in detail.
Based on the results presented above, the
following main conclusions can be formulated:
Damping coefficient
c
d
,
Fractional Kelvin
model
Fractional Maxwell
model
Storey
Sequential
method
PSO
method
Sequential
method
PSO
method
1
0
0
0
0.78
2
0
0
0
0.78
3
100.0
87.57
0
0.78
4
50.0
47.68
0
0.78
5
100.0
106.18
0
0.78
6
50.0
56.68
0
0.78
7
100.0
120.77
350.0
347.23
8
50.0
26.68
0
0.78
9
50.0
54.45
150.0
146.48
10
0
0
0
0.78
Total
500.0
500.01
500.0
499.95
ized using Equation (7), therefore, the values
given in Table 7 differ from
c
min
=
1
. It justifies,
for example, that it is possible to find, using the
sequential optimization method, a solution which
is near the global optimum of the considered
optimization problem.
The Maple program was used to obtain all of
the results presented in this example.
•
The problem of optimal distribution of VE
dampers modelled using the rheological
models with fractional derivative or using
the generalized classical rheological mod-
els is solved for the first time.
CONCLUDING REMARKS AND
FUTURE RESEARCH WORKS
•
The results presented prove the effective-
ness and applicability of the proposed
approach.
The problems of the optimal location of VE
dampers on the planar frame structures and de-
termination of the optimal values of parameters
of dampers are considered in this chapter. VE
dampers are modelled using several rheological
models, i.e., the generalized Kelvin and Maxwell
models, both with seven parameters, and the
three-parameter Kelvin and Maxwell models with
fractional derivatives. The mathematical formula-
tion for structures with VE dampers, modelled by
•
The optimal distribution of dampers within
a structure strongly depends on the ad-
opted objective function and the structure
characteristics.
•
The results of optimization of problems in
which VE dampers are modelled using the
generalized Kelvin model and the general-
ized Maxwell model are almost identical
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